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Anamorphosis of 3D-Objects & 3D Printing

Posted 7 years ago

2D cylindrical mirror anamorphosis is well known and in the past, I made several Wolfram Demonstrations on the subject. Such as : Cylindrical Mirror Anamorphosis and Cylindrical Anamorphosis of Some Popular Images,

George Beck of Wolfram Research drew my attention to the website of the London artist Jonty Hurwitz: Wild New Anamorphic Sculptures From the Warped Mind of Jonty Hurwitz. And I went on to see what could be done using the Graphic capabilities of Mathematica...

enter image description here

These sculptures are cases of 3D mirror or catoptric anamorphosis in contrast with its counterpart, perspective or oblique anamorphosis .

We first need to look at the geometry of the setup for cylindrical mirror anamorphosis: A deformed 3D image (sculpture) is resting on the x-y plane, outside a right cylindrical mirror, this is the anamorphic image. R is a point of this anamorphic image. A viewer looks at the cylindrical mirror and sees a reflected, realistic looking sculpture 'inside' the cylinder, this is the reflected image. I is a point of this reflected image A point R of the anamorphic image will be reflected by the mirror toward the observer's eye, located at V . The light path will form equal angles with the normal vector n, perpendicular at the point Q on the cylinder. Q is the intersection with the cylinder of the view line connecting V and I.


The viewer at V perceives the anamorphic points R as points I, the artist has to create anamorphic points R from the reflected points I. Based on three observations, a function (anamorphCyl3D) can be made that maps the reflected image points [ScriptCapitalI] into their anamorphic match R:

  1. both I and R are in the same horizontal plane: zi=zr
  2. the lines VQ and RQ form equal angles with the normal n at Q according to the law of reflection
  3. the distances IQ and RQ are equal per the same law of reflection

    anamorphCyl3D[(*image point position*)
      iPt : {xi_, yi_, zi_},(*viewpoint*)vPt : {0, yv_, zv_}] :=
     Quiet@Module[{t, R1, R2, hi = 5, qPt, nV, aV, rV, vV, solt},
       (*view line*)R1 = InfiniteLine[{iPt, vPt}];
       (*cylinder hull: mirror*)
       R2 = RegionBoundary[Cylinder[{{0., 0, 0}, {0, 0, 10}}, 1.]];
       (*intersection point nearest to viewpoint*)
       qPt = Nearest[{x, y, z} /. 
           NSolve[{x, y, z} \[Element] R1 && {x, y, z} \[Element] R2, {x, 
             y, z}, Reals], vPt][[1]];
       (*view vector*)vV = vPt - iPt;
       (*normal vector at qPt*)nV = {qPt[[1]], qPt[[2]], 0};
       (*vertical dropdown from vPt*)aV = Projection[vV, nV] - vV;
       (*reflection vector of vPt from iPt*)rV = vV + 2 aV;
       t = -(qPt[[3]] - iPt[[3]])/rV[[3]];
       (*anamorphic point*)t rV + qPt]

This function can be simplified and compiled to increase the speed. This is more than necessary since we will have to process hundreds, if not thousands of points:

anamorphCyl3DCF = 
 Compile[{{xi, _Real}, {yi, _Real}, {zi, _Real}, {yv, _Real}},
  Module[{t1, t2},
   t1 = Sqrt[(yi - yv)^2 - xi^2 (-1 + yv^2)]; 
   t2 = 1/(xi^2 + (yi - 
         yv)^2); {-t2^2 xi ((xi^2 + (yi - yv)^2) ((yi - yv) yv + 
          t1) - (xi^2 + yi^2 - yi yv + 
          t1) (xi^2 (-1 + 2 yv^2) - (yi - yv) (yi - yv + 2 yv t1))), 
    t2 (xi^2 yv + (-yi + yv) t1 + 
       t2 (xi^2 + yi^2 - yi yv + t1) (-(yi - yv)^3 + 
          xi^2 (-yi + yv + 2 yi yv^2 - 2 yv^3 + 2 yv t1))), zi}], 
  CompilationTarget -> "C", Parallelization -> True, 
  RuntimeOptions -> "Speed"]

Let's try this first by applying the function to points on the surface of an object defined by ListSurfacePlot, e.g a sphere:

sphere =(*points on the spere surface*)
  Flatten[Table[.9 {Cos[\[Phi]] Sin[\[Theta]], 
      Sin[\[Phi]] Sin[\[Theta]], 1. - Cos[\[Theta]]}, {\[Phi], 0., 
     2 \[Pi], \[Pi]/16}, {\[Theta], 0., \[Pi], \[Pi]/16}], 1];
anamorphicSphere =(*their anamorphic counterpart*)

  Map[anamorphCyl3DCF @@ Flatten[{#, 6.}] &, sphere];
     Cylinder[{{0, 0, 0}, {0, 0, 2}}, 1]}, 
    ListSurfacePlot3D[sphere, Mesh -> All][[1]]}, Axes -> True],
  ListSurfacePlot3D[anamorphicSphere, Mesh -> All, 
   PerformanceGoal -> "Quality", PlotTheme -> "ThickSurface"]}, 
 ImageSize -> 350]


Or we could apply the function directly inside a ParametricPlot3D object, e.g. a torus

GraphicsRow[{ParametricPlot3D[.33 {(2 + Cos[v]) Cos[u], Sin[v], 
     3. + (2 + Cos[v]) Sin[u]}, {u, 0, 2. \[Pi]}, {v, 0, 2. \[Pi]}],
  ParametricPlot3D[ anamorphCyl3DCF @@ Flatten[{#, 6.}] &[.33 {(2 + Cos[v]) Cos[u], 
      Sin[v], 3. + (2 + Cos[v]) Sin[u]}], {u, 0, 2. \[Pi]}, {v, 0, 2. \[Pi]}]}, ImageSize -> 350]

enter image description here

Here we see both the reflected torus and its anamorphic counterpart together with the cylindric mirror. One can rotate the torus along one of its main axes or change the distance of the observer's eye from the center of the mirror.

  torus = RotationMatrix[\[Theta], 
     Switch[dir, 1, {1, 0, 0}, 
      2, {0, 0, 1}]].   (.33 {Cos[u] (2 + Cos[v]), 
       Sin[v], (2 + Cos[v]) Sin[u]});
     ParametricPlot3D[torus, {u, 0, 2. \[Pi]}, {v, 0, 2. \[Pi]}, 
      Mesh -> 10, PlotPoints -> 25],
      anamorphCyl3DCF @@ Flatten[{#, yv}] &[torus], {u, 0, 
       2. \[Pi]}, {v, 0, 2. \[Pi]}, Mesh -> 10, PlotPoints -> 36],
     Graphics3D[{Cylinder[{{0, 0, -1}, {0, 0, -.99}}, 5], 
       Opacity[.35], Cylinder[{{0, 0, -1}, {0, 0, 2}}, 1]}]}, 
    PlotRange -> {{-2, 2}, {-1, 3.5}, {-1, 2}}, Axes -> False, 
    Boxed -> False, ViewPoint -> {0, 2., 3.1}, ImageSize -> 250]],
 "eye distance", {{yv, 6., ""}, 2, 24, ImageSize -> Small},
 "\nrotation angle", {{\[Theta], 0., ""}, -\[Pi], \[Pi], 
  ImageSize -> Small}, 
 Row[{"\nrotation axis", 
   Control[{{dir, 1, ""}, {1 -> "x-axis", 2 -> "z-axis"}}]}], 
 ControlPlacement -> Left, SynchronousUpdating -> True]


We can now also apply this function to the vertices of a polygonal mesh of a discretized object. For example, the ones from ExampleDtata, "Geometry3D" in the Wolfram Language [1]. We first need to scale the coordinates within the limits of a right cylinder with radius 1, centered around the z-axis:

getAndRescale[example_String] := 
 Module[{data, ranges, maxRange, temp1, temp2},
  data = ExampleData[{"Geometry3D", example}, "PolygonObjects"];
  ranges = MinMax@Flatten[data[[All, 1]], 1][[All, #]] & /@ Range[3];
  maxRange = MaximalBy[Most@ranges, Abs[Subtract @@ #] &]; 
  temp1 = MapAt[Rescale[#, First@maxRange, {-1., 1.}] &, 
    data, {All, 1, All, 1}];
     Last@ranges, {0, 
      2*Subtract @@ ranges[[3]]/Subtract @@ ranges[[1]]}] &, 
   temp1, {All, 1, All, 3}]]

Applied to the "UtahTeapot":

Module[{yv = 6., data, anaData},
 data = getAndRescale["UtahTeapot"];
 anaData = Map[anamorphCyl3DCF @@ Flatten[{.96 #, yv}] &, data, {3}];
 g = Graphics3D[anaData];
   anaData, {LightGray, 
    Cylinder[{{0, 0, 0}, {0, 0, .01}}, 10]}, {Opacity[.35], 
    Cylinder[{{0, 0, 0}, {0, 0, 2}}, 1]}}, 
  PlotRange -> {{-2, 2}, {-1, 3}, {0, 2}}, 
  ViewPoint -> {2.65, 1.6, 1.45}]]

Utah teapot

The anamorphic teapot can now be printed using Printout3D and uploaded to an online 3D printing service as e.g. 3DHubs:

Printout3D[g, "3DHubs", RegionSize -> Quantity[8, "Centimeters"]]

teapot from hub

Afterwards, this can be tested in the reflection of a real homemade cylindrical mirror [2]:

teapot printed

Or we can try the "Galleon" saved as an STL file with Printout3D:


enter image description here

And we could try the famous Stanford Bunny with almost 70k polygons uploaded with Printout3D to 3D printer Sculpteo:


Any more artists out there that want to try their skills at 3D anamorphic printing using Mathematica?


[1] By mapping the vertices of polygonal meshes to their anamorphic counterpart and drawing a straight line between the vertices, we get the same limitation a with the original: only an infinite number of vertices will give the absolute correct object. But this is hardly a limitation to get an interesting result...

[2] Cardboard cylinder from a discarded salt dispenser. Reflective window film glued around it. Diameter 60mm.

POSTED BY: Erik Mahieu
8 Replies
Posted 7 years ago

Best is to extract the Graphics3D or ParemtricPlot3D data as generated of the code and use Printout3D[data,"filename.stl"] to save the model as a 3D print file.

POSTED BY: Erik Mahieu


Can the resulting 3D models be downloaded somewhere? Or can I run the code online (Without owning Mathematica) ?



A Utah teapot aficionado.

POSTED BY: Remko Siemerink
Posted 7 years ago

Could be a next project:"AMBIGUOUS CYLINDERS". Again a nice combination of math, art and imagination. Thanks for sharing.

POSTED BY: Erik Mahieu

What a great post.

I'm wondering if it might be possible to adapt the concept and use Mathematica to generate objects with interesting reflecting properties, like this: Ambiguous Garage Roof by Kokichi Sugihara

.... which could then be 3D printed.

POSTED BY: Jonathan Kinlay
Posted 7 years ago

@Erik Mahieu - that's impressive :-) Well done you cracked my code. Lets hook up and break some new boundaries together! Let's make some art!

Kind Regards Jonty

POSTED BY: Jonty Hurwitz

enter image description here - Congratulations! This post is now a Staff Pick as distinguished on your profile! Thank you for your wonderful contributions. Please, keep them coming! (We have also corrected the Printout3D type you indicated in your comment).

POSTED BY: Moderation Team
Posted 7 years ago

Glad you enjoyed it. Anamorphism is really an exciting subject with an intriguing combination of mathematics, imagination and art.

The anamorphic transformation code I took from my former demonstrations. After I discovered the new Printout3D, I got really interested in 3D printing and it uploads automatically to e.g. 3DHubs. The advantage of 3DHubs is that it is some type of 3D printer sharing service. It connects to people in your area who have excess printing capacity and try to sell it. So I found an industrial design student at 3 km from my home and the guy printed it overnight. So the whole "enterprise" took no more than a couple of days. With thanks to the power of Mathematica!

If somebody is interested, I am willing to help and we could make a larger scale or more sophisticated object to demonstrate an example of something like: "from code to 3D art in one day with Mathematica"! Could make an exclusive conversation piece for any math geek's office...

POSTED BY: Erik Mahieu

@Erik Mahieu this absolutely wonderful! How long did this project took to complete, including coding, 3D printing, ordering from 3DHubs, etc.?

POSTED BY: Vitaliy Kaurov
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